Solution set

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In mathematics, a solution set is a set of possible values that a variable can take on in order to satisfy a given set of conditions (which may include equations and inequalities).

Formally, for a collection of polynomials <math>\{f_i\}</math> over some ring <math>R</math>, a solution set is defined to be the set <math>\{x\in R:\forall i\in I, f_i(x)=0\}</math>.

1 Examples
2 Remarks
3 Other meanings
- 3.1 Examples
4 See also

[edit] Examples

1. The solution set of <math>f(x):=x</math> over the real numbers is the set {0}.

2. For any non-zero polynomial <math>f</math> over the complex numbers in one variable, the solution set is made up of finitely many points. However, for a complex polynomial in more than one variable the solution set has no isolated points.

[edit] Remarks

In algebraic geometry solution sets are used to define the Zariski topology. See affine varieties.

[edit] Other meanings

More generally, the solution set to an arbitrary collection E of relations (E_i) (i varying in some index set I) for a collection of unknowns <math>{(x_j)}_{j\in J}</math>, supposed to take values in respective spaces <math>{(X_j)}_{j\in J}</math>, is the set S of all solutions to the relations E, where a solution <math>x^{(k)}</math> is a family of values <math>{(x^{(k)}_j)}_{j\in J}\in \prod_{j\in J} X_j</math> such that substituting <math>{(x_j)}_{j\in J}</math> by <math>x^{(k)}</math> in the collection E makes all relations "true".

(Instead of relations depending on unknowns, one should speak more correctly of predicates, the collection E is their logical conjunction, and the solution set is the inverse image of the boolean value true by the associated boolean-valued function.)

The above meaning is a special case of this one, if the set of polynomials f_i if interpreted as the set of equations f_i(x)=0.

[edit] Examples

The solution set for E = { x+y = 0 } w.r.t. <math>(x,y)\in\mathbb R^2</math> is S = { (a,-a) ; a ∈ R } .
The solution set for E = { x+y = 0 } w.r.t. <math>x\in\mathbb R</math> is S = { -y } . (Here, y is not "declared" as an unknown, and thus to be seen as a parameter on which the equation, and therefore the solution set, depends.)
The solution set for <math> E = \{ \sqrt x \le 4 \} </math> w.r.t. <math>x\in\mathbb R</math> is the interval S = [0,2] (since the equation (inequality) is not well defined for negative numbers).
The solution set for <math> E = \{ \exp(i x) = 1 \} </math> w.r.t. <math>x\in\mathbb C</math> is S = 2 π Z (see Euler's identity).